The invention relates generally to the field of analog-to-digital (xe2x80x9cA/Dxe2x80x9d) or digital-to-analog (xe2x80x9cD/Axe2x80x9d) signal conversion. More particularly, the invention relates to a system and method for digitizing the amplitude of a stream of digitized samples using a one-bit 2nd order sigma delta modulator (xe2x80x9cxcexa3xcex94 modulationxe2x80x9d).
In a typical analog to digital signal conversion, analog signals (xe2x80x9cx(t)xe2x80x9d) are sampled and then transformed into digital signal (xe2x80x9cx{circumflex over ( )}kxe2x80x9d) equivalents. Analog signals, x(t), are both continuous in time and in amplitude. In order to properly digitize an analog signal, the analog signal""s amplitude and time must be digitized. Sigma delta modulators digitize the analog signal""s amplitude.
The maximum frequency for an analog signal, x(t), is known as fmax. Analog signals include, but are not limited to, audio signals. An exemplary audio signal fmax=20,000 Hertz. The rate at which analog signals are sampled is known as fsamp. Once an analog signal has been sampled, the original signal which was both continuous in time and amplitude is now represented as a set of numbers (xe2x80x9cxkxe2x80x9d) which is no longer continuous in time. The set of numbers, xk, is represented as
xk=x(k/fsamp)xe2x80x83xe2x80x83(Equation 1)
where k=0, 1, 2, 3 and fsamp=constant.
It is well known in the art that if
fsamp greater than 2fmaxxe2x80x83xe2x80x83(Equation 2)
it is possible to recover the original analog x(t) signal from the signal set, xk. After the analog signal, x(t), has been sampled and converted into the sequence of numbers xk, the signal is feed into a xcexa3xcex94 modulator which digitizes the amplitude of the signal set xk. The xcexa3xcex94 modulator produces an output qk which is a sequence of samples that is both discrete in time and in amplitude. Because the xcexa3xcex94 modulator distorts the xk sequence, qk is fed into a lowpass filter. If a xcexa3xcex94 modulator is properly designed, a good approximation of xk can be recovered from qk. This is accomplished by lowpass filtering the qk signal with a cutoff frequency (xe2x80x9cfcutoffxe2x80x9d) equal to fmax. A xcexa3xcex94 modulator""s performance is measured by evaluating the time-average of the squared error (x{circumflex over ( )}kxe2x88x92xk)2, called the xe2x80x9cmean squared errorxe2x80x9d (xe2x80x9cMSExe2x80x9d). In practice, the MSE results in the perception of noise. Therefore, to avoid additional signal imperfections, the MSE should be minimized as much as possible.
There are various types of xcexa3xcex94 modulators which approximate and digitize a signal""s amplitude. One bit xcexa3xcex94 modulators, output two discrete levels of signals, while xe2x80x9cmulti-bitxe2x80x9d xcexa3xcex94 modulators output more than two discrete levels of signals. Theoretically, multi-bit xcexa3xcex94 modulators out-perform one-bit xcexa3xcex94 modulators. In other words, theoretically, the MSE in multi-bit xcexa3xcex94 modulators is less than the MSE in one bit xcexa3xcex94 modulators. However, practically speaking, multi-bit xcexa3xcex94 modulators have complex circuit designs and are sensitive to analog circuit imperfections. xe2x80x9cNth orderxe2x80x9d xcexa3xcex94 modulators include nth order integrators. Once again, the higher the order, the higher the theoretical performance, however, high order xcexa3xcex94 modulators, when implemented, have complex circuit designs. The preferred embodiment of the present invention improves the one-bit 2nd order xcexa3xcex94 modulator.
Prior art versions of xcexa3xcex94 modulators were primarily linear and time-invariable. The only exception to the linear constraint was quantization, which was non-linear. Consequently, besides quantization, the building blocks of prior art xcexa3xcex94 modulators were limited to linear functions such as delay operators (xe2x80x9cDxe2x80x9d), constant coefficient multipliers, and signal additions.
Nonlinear circuits add errors to input error signals which human ears detect and can be annoyed by. These error signals are called xe2x80x9charmonics.xe2x80x9d A/D conversion requires nonlinear quantization. However, because quantization is incorporated in the feedback process of a xcexa3xcex94 modulator, the error that the quantizer generates becomes background noise. Human ears are less sensitive to this background noise. The present invention incorporates this concept.
In conclusion, prior art versions of one-bit 2nd order xcexa3xcex94 modulators perform with an MSE decay rate of 12 dB/octave, while prior art versions of multi-bit 2nd order xcexa3xcex94 modulators achieve the preferred 15 dB/octave MSE decay rate. However, multi-bit xcexa3xcex94 modulators are more sensitive to analog circuit imperfection and are complex to construct.
What is needed in the art is an improved one-bit xcexa3xcex94 modulator, which the present invention provides.
These and other problems are addressed by the xcexa3xcex94 modulator of the present invention. The present invention utilizes the understanding that nonlinear operations incorporated into the feedback process of a xcexa3xcex94 modulator generate background noise which is generally acceptable to human ears. The present invention incorporates this understanding in the circuit design of the present invention to emulate a near ideal MSE decay rate. The present invention expands the acceptance of non-linear operations in the xcexa3xcex94 modulator""s feedback loop by incorporating circuitry to implement the following equations in the xcexa3xcex94 modulator""s vector quantizer:
Q(xk, ukxe2x88x921, vkxe2x88x921)=qk[(7xe2x88x923|xk|)x+(6xe2x88x924|xk|)ukxe2x88x921+(4xe2x88x928)|xk|vkxe2x88x921+sign(xk) (4(xk+ukxe2x88x921)2xe2x88x921/12)]xe2x80x83xe2x80x83(Equation 3)
Q(xk, ukxe2x88x921, vkxe2x88x921)=qk[(10xe2x88x924|xk|)xk+(9xe2x88x926|xk|)ukxe2x88x921+(6xe2x88x9212|xk|)vkxe2x88x921]xe2x80x83xe2x80x83(Equation 4)
Q(xk, ukxe2x88x921, vkxe2x88x921)=qk[(14bxe2x88x923|xk|)x+(12bxe2x88x924|xk|)ukxe2x88x921+(8bxe2x88x928)|xk|vkxe2x88x921+sign(xk) (4(xk+ukxe2x88x921)2xe2x88x92b2/3)]xe2x80x83xe2x80x83(Equation 5)
Q(xk, ukxe2x88x921, vkxe2x88x921)=qk[(20bxe2x88x924|xk|)xk+(18bxe2x88x926|xk|)ukxe2x88x921+(12bxe2x88x9212|xk|)vkxe2x88x921]xe2x80x83xe2x80x83(Equation 6)
The xcexa3xcex94 modulator of the present invention reduces circuit complexity by outputting only two discrete signal levels. Consequently, the 2nd order xcexa3xcex94 modulator of the present invention is a one-bit modulator design. Accordingly, the circuitry involved with the xcexa3xcex94 modulator of the present invention is less complex and is less susceptible to analog circuit imperfections. In addition, the xcexa3xcex94 present invention achieves the MSE decay rate of prior art multi-bit modulators using only one bit. Thus, the present invention achieves the ideal MSE decay rate of 15 dB/octave without the complex circuitry and increased sensitivity to circuit imperfections when using multi-bits.
As mentioned above, the MSE decay of 12 dB/octave is the limit for prior art one-bit 2nd order xcexa3xcex94 modulators. This is because the only non-linear operation in prior art xcexa3xcex94 modulators was quantization. A linear operation has 3 forms: multiplication by a constant, additions of signals, or delay of a signal. The present invention is a new type of xcexa3xcex94 modulator in which both the quantizer and the vector quantizer (xe2x80x9cQxe2x80x9d), that includes the quantizer, is non-linear. A quantizer is represented by a comparator in xcexa3xcex94 modulator designs. The xcexa3xcex94 modulator, according to the present invention, departs from the prior art theory that the quantizer can be the only non-linear function in a xcexa3xcex94 modulator. In its novel construction, the present invention includes a vector quantizer that breaks the 12 dB/oct barrier of known prior art one-bit 2nd order xcexa3xcex94 modulators. A vector quantizer of the preferred embodiment includes both the standard quantizer as well as has additional non-linear circuitry.
The xcexa3xcex94 modulator of the present invention has a vector quantizer, Q, which is memoriless. Memoriless means that, at every instant, the outputs of the vector quantizer, Q, are a function of only the instantaneous values of the inputs, and not a function of past input values. The vector quantizer, Q, of the present invention is a xe2x80x9cmemorilessxe2x80x9d 3-input function that outputs binary numbers at two discrete levels. Prior art xcexa3xcex94 modulators have constrained quantizers where the output is multiplied by the sign of a linear combination of its inputs. The output of the vector quantizer of the present invention, Q(xk, ukxe2x88x921, vkxe2x88x921), is multiplied by the quantization of a polynomial function to the degree necessary in which to achieve the ideal MSE decay rate. This amounts to quantizing a 1st degree polynomial function of the inputs. This MSE decay rate can be achieved using a polynomial to the 2nd degree. Mathematical derivations lead to the following vector quantizer functions:
Q(xk, ukxe2x88x921, vkxe2x88x921)=qk[(7xe2x88x923|xk|)xk+(6xe2x88x924|xk|)ukxe2x88x921+(4xe2x88x928|xk|)vkxe2x88x921+sign(xk) (4(xk+ukxe2x88x921)2xe2x88x921/12)]xe2x80x83xe2x80x83(Equation 3)
Q(xk, ukxe2x88x921, vkxe2x88x921)=qk[(10xe2x88x924|xk|)xk+(9xe2x88x926|xk|)ukxe2x88x921+(6xe2x88x9212|xkxe2x88x921)vkxe2x88x921]xe2x80x83xe2x80x83(Equation 4)
Q(xk, ukxe2x88x921, vkxe2x88x921)=qk[(14bxe2x88x923|xk|)x+(12bxe2x88x924|xk|)ukxe2x88x921+(8bxe2x88x928)|xk|vkxe2x88x921+sign(xk) (4(xk+ukxe2x88x921)2xe2x88x92b2/3)]xe2x80x83xe2x80x83(Equation 5)
Q(xk, ukxe2x88x921, vkxe2x88x921)=qk[(20bxe2x88x924|xk|)xk+(18bxe2x88x926|xk|)ukxe2x88x921+(12bxe2x88x9212|xk|)vkxe2x88x921]xe2x80x83xe2x80x83(Equation 6)
Were there any further reduction of the quadratic functions complexity, the polynomial function would only be to the 1st degree. This would turn the one bit 2nd order xcexa3xcex94 modulator of the present invention into a linearly constrained circuit of prior art xcexa3xcex94 modulators. Thus, the polynomial function of the quantizer according to the invention is non-linear and can achieve an MSE decay rate of up to 15 dB/octave.